In the low Mach number limit, terms proportional to \( 1/Ma^2 \)  in equations (REF )
yield stiff components of the resulting semidiscretized ODE system. Therefore, following
as remarked above [1]}, [2]}, it is appropriate to couple implicitly
the energy equation to the momentum equation, while the continuity equation can be discretized in a fully explicit fashion.
While this would be sufficient to yield an efficient time discretization approach for the purely hyperbolic system associated to (REF ) in absence of gravity, in regimes for which
\(Pr\approx O(1), \ \ \ Fr <<1 \) 
